Classical scaling dimension
Encyclopedia
In theoretical physics
, namely quantum field theory
, the classical scaling dimension of an operator O is the power of mass
of an operator determined by dimensional analysis
from the Lagrangian
(1 for elementary boson
ic fields including the vector potentials, 3/2 for elementary fermion
ic fields etc.). It is also called the naïve dimension or the engineering dimension.
The dimension of a product is the sum of the dimensions. For example, in four-dimensional quantum field theory
, the Lagrangian
must have dimension of mass to the fourth power. By a dimension of the individual terms, we mean the dimension of the corresponding operators with the coefficients removed.
The operators whose coefficients are dimensionless are called marginal operators in the context of renormalization group
or dimension four operators. The operators of dimension five are dimension five operators and much like other operators whose dimension exceeds four, they are irrelevant operators. Operators with dimensions lower than four, e.g. dimension three operators, are relevant operators.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, namely quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, the classical scaling dimension of an operator O is the power of mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
of an operator determined by dimensional analysis
Dimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
from the Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
(1 for elementary boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
ic fields including the vector potentials, 3/2 for elementary fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
ic fields etc.). It is also called the naïve dimension or the engineering dimension.
The dimension of a product is the sum of the dimensions. For example, in four-dimensional quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, the Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
must have dimension of mass to the fourth power. By a dimension of the individual terms, we mean the dimension of the corresponding operators with the coefficients removed.
The operators whose coefficients are dimensionless are called marginal operators in the context of renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
or dimension four operators. The operators of dimension five are dimension five operators and much like other operators whose dimension exceeds four, they are irrelevant operators. Operators with dimensions lower than four, e.g. dimension three operators, are relevant operators.
See also
- Conformal weight
- Anomalous scaling dimensionAnomalous scaling dimensionIn theoretical physics, by anomaly one usually means that the symmetry remains broken when the symmetry-breaking factor goes to zero.When the symmetry which is broken is scale invariance, then true power laws usually cannot be found from dimensional reasoning like in turbulence or quantum field...
- Dimensional analysisDimensional analysisIn physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...